Distributed single-snapshot direction of arrival estimation method based on two-dimensional Hankel matrix completion

Abstract

The application relates to the technical field of array signal processing, in particular to a distributed single-shot direction of arrival estimation method based on two-dimensional Hankel matrix completion, which comprises the following steps: acquiring signals received by each sensor of each subarray and stacking the signals into a global data matrix, obtaining an under-sampling observation signal containing additive noise based on an element sampling operator and the global data matrix; constructing a two-dimensional Hankel matrix based on the global data matrix; calculating a local cost function based on the under-sampling observation signal, converting the completion problem of the two-dimensional Hankel matrix into a distributed consistency optimization problem, realizing consistency iteration through a DHEX algorithm, obtaining a global low-rank solution, and then obtaining a reconstructed signal; and processing the reconstructed signal by using a two-dimensional subspace method to obtain a direction of arrival estimation value. The method can realize high-precision and scalable direction of arrival estimation under the condition of only local incomplete observation.

Description

TECHNICAL FIELD

[0001] Embodiments of the present application relate to the technical field of array signal processing, in particular to a distributed single snapshot direction of arrival estimation method based on two-dimensional Hankel matrix completion. BACKGROUND

[0002] The direction of arrival (DOA) estimation in array signal processing is one of the basic problems in wireless communication, radar and sonar systems. Current DOA estimation mostly relies on subspace methods, including the spatial spectrum estimation algorithm (MUSIC) and the rotation invariant subspace algorithm (ESPRIT), which can achieve high resolution DOA estimation under the conditions of multiple snapshots and complete data. These classic methods need to be based on some ideal assumptions, such as multiple snapshot acquisition, complete array observation data, and centralized processing architecture. However, in practical applications, these assumptions are often difficult to meet, especially under the conditions of single snapshot acquisition, incomplete sampling, and communication constraints.

[0003] For the single snapshot and incomplete sampling problem, existing research teams have proposed to promote sparse or incomplete array observations to Hankel matrices and utilize their low-rank structure to complete the matrix in a centralized framework, thereby recovering the complete data for DOA estimation. However, this method relies on centralized data fusion and is difficult to adapt to distributed array environments, with the drawbacks of heavy communication burden, insufficient scalability, and unsuitability for real-time applications.

[0004] With the development of array signal processing technology, distributed DOA estimation methods are implemented through consistency algorithms or local cooperation without centralized fusion, which has improved communication efficiency and scalability to some extent. However, under the conditions of single snapshot acquisition, random incomplete sampling, and communication constraints, the current distributed methods still have the following shortcomings.

[0005] First, the estimation accuracy is insufficient under single snapshot conditions. Due to the inability to construct a stable sample covariance matrix, traditional subspace methods fail, parameter distinguishability decreases, and noise significantly affects the estimation results.

[0006] Second, incomplete sampling leads to completion difficulties. The method of data completion using the low-rank property of Hankel matrices is usually based on a centralized architecture. In a distributed situation, each subarray only has local observations, making it difficult to ensure that the low-rank property of the global Hankel matrix can be effectively utilized, thereby leading to insufficient completion accuracy. SUMMARY

[0007] In order to solve the above technical problems, the embodiment of the present application proposes a distributed single-snapshot direction of arrival estimation method based on two-dimensional Hankel matrix completion, a distributed processing framework combining two-dimensional Hankel matrix low-rank completion and subspace estimation is constructed, high-precision and scalable DOA estimation can be realized without centralized data fusion, the communication and calculation overheads caused by centralized processing are effectively reduced, and the consistency and convergence of distributed processing are well ensured.

[0008] In order to achieve the above purpose, the embodiment of the present application proposes a distributed single-snapshot direction of arrival estimation method based on two-dimensional Hankel matrix completion, which is suitable for a distributed linear array system composed of multiple sub-arrays, and the method comprises the following steps: acquiring signals received by each sensor of each sub-array and stacking the signals into a global data matrix, obtaining an under-sampling observation signal containing additive noise based on an element-wise sampling operator and the global data matrix; performing two-dimensional Hankel low-rank promotion on the global data matrix to construct a two-dimensional Hankel matrix; calculating a local cost function based on the under-sampling observation signal, converting the completion problem of the two-dimensional Hankel matrix into a distributed consistency optimization problem, realizing consistency iteration through a DHEX algorithm, obtaining a global low-rank solution and completing the reconstruction of a complete signal to obtain a reconstructed signal; and processing the reconstructed signal by using a two-dimensional subspace method to obtain a direction of arrival estimation value.

[0009] In order to achieve the above purpose, the embodiment of the present application further proposes a distributed single-snapshot direction of arrival estimation device based on two-dimensional Hankel matrix completion, which is suitable for a distributed linear array system composed of multiple sub-arrays, and the device comprises the following modules: an under-sampling observation signal acquisition module, which is used to acquire signals received by each sensor of each sub-array and stack the signals into a global data matrix, and obtain an under-sampling observation signal containing additive noise based on an element-wise sampling operator and the global data matrix; a two-dimensional Hankel low-rank promotion module, which is used to perform two-dimensional Hankel low-rank promotion on the global data matrix to construct a two-dimensional Hankel matrix; a signal reconstruction module, which is used to calculate a local cost function based on the under-sampling observation signal, convert the completion problem of the two-dimensional Hankel matrix into a distributed consistency optimization problem, realize consistency iteration through a DHEX algorithm, obtain a global low-rank solution and complete the reconstruction of a complete signal to obtain a reconstructed signal; and an estimation execution module, which is used to process the reconstructed signal by using a two-dimensional subspace method to obtain a direction of arrival estimation value.

[0010] To achieve the above objectives, embodiments of this application also propose an electronic device, including a processor and a memory, wherein the memory stores instructions executable by the processor, and the processor is configured to execute the instructions such that the electronic device can implement the distributed single-snapshot direction-of-arrival estimation method based on two-dimensional Hankel matrix completion as described above.

[0011] To achieve the above objectives, embodiments of this application also propose a computer-readable storage medium storing a computer program that, when executed by a processor, enables a distributed single-snapshot direction-of-arrival estimation method based on two-dimensional Hankel matrix completion as described above.

[0012] Optionally, the distributed linear array system is composed of It consists of uniform linear subarrays, each subarray containing One sensor, and All are integers greater than 1, and the element spacing is denoted as . The displacements of adjacent subarrays are denoted as , No. The first of the sub-arrays The position of each sensor is expressed by the formula:

[0013] ;

[0014] in, For the first The first of the sub-arrays The location of each sensor ， ;

[0015] The signals received by each sensor in each subarray are acquired and stacked into a global data matrix, including:

[0016] Assume there are a total of A narrowband far-field wave source, denoted as the... A narrowband far-field wave source has a global complex amplitude. The global complex amplitude is determined solely by the narrowband far-field wave source itself and remains consistent across all subarrays;

[0017] In the absence of noise, the first The first of the sub-arrays The signal received by each sensor can be expressed by the formula:

[0018] ;

[0019] in, For the first The first of the sub-arrays signals received by the sensors, for the wavelength, denotes the DOA of the unknown th narrowband far-field wave source, denotes the imaginary unit;

[0020] stacks all the signals received by the th subarray into an observation vector , stacks all the observation vectors corresponding to the th subarray into a global data matrix ;

[0021] wherein, , the upper right corner denotes the transpose operation, , , denotes the complex field of dimension .

[0022] Optionally, define as the observation index set, , define as the element-wise sampling operator, which is expressed by the formula:

[0023] ;

[0024] wherein, denotes the iteration variable, denotes the local estimation value of the th subarray for the th sensor pair ;

[0025] Based on the element-wise sampling operator and the global data matrix, the undersampled observation signal containing additive noise is obtained by the following formula:

[0026] ;

[0027] wherein, denotes the additive noise, denotes the undersampled observation signal, , ;

[0028] define as the local observation index set of the th subarray, , , denotes the identity, and dimension consistent, but only the first entries of the corresponding row are kept, defining a local element-wise sampling operator for the th subarray, which is expressed as

[0029] ;

[0030] a local under-sampled observation signal for the th subarray, which is expressed as

[0031] ;

[0032] where denotes that the th subarray can only access the local under-sampled observation signal, and the th subarray can only access .

[0033] Optionally, the element in the th row and the th column of the global data matrix satisfies

[0034] ;

[0035] where , , ;

[0036] Based on this, the Vandermonde vector and are expressed as

[0037] ;

[0038] ;

[0039] where , , denotes a complex field with dimension , denotes a complex field with dimension ;

[0040] Based on and , the global data matrix is expressed as , , denotes the rank of ;​

[0041] Two-dimensional Hankel low-rank promotion is performed on the global data matrix to construct a two-dimensional Hankel matrix, including:

[0042] Taking an integer , , , , , ;

[0043] A one-dimensional Hankel operator along the sensor dimension is defined , , , where is a complex field with dimension , and , satisfies ;

[0044] The global data matrix is applied row by row , obtaining small block matrices, and the small block matrix corresponding to the th row of the global data matrix is , ;

[0045] Based on all the small block matrices, a two-dimensional Hankel matrix is constructed along the matrix dimension, denoted as:

[0046] ;

[0047] where , is a complex field with dimension .

[0048] Optionally, based on the under-sampled observation signal, the calculation of the local cost function converts the completion problem of the two-dimensional Hankel matrix into a distributed consistency optimization problem, including:

[0049] Communication modeling is performed, and the communication network between the subarrays is modeled as an undirected graph , , where is a set of subarrays, is a communication edge, , and each subarray can only exchange data with its neighbor subarrays in the optimization process to meet the distributed constraint;

[0050] Distributed modeling based on two-dimensional Hankel matrix completion is performed, and and is odd, let and satisfy the constraint: , ;

[0051] lifting two-dimensional Hankel matrices is a square matrix , , denotes the complex field of dimension ; ;

[0052] By using the symmetric low-rank decomposition method, decompose into parameterized , , denotes the complex field of dimension ;

[0053] Define the local cost function of the th submatrix as:

[0054] ;

[0055] where denotes the local cost of the th submatrix, denotes the local estimate of the th submatrix to , is the sampling rate, , is the identity operator, , is the adjoint of , is the invertible linear weighting operator, and the upper right superscript denotes the inverse operation, for , satisfies , and are and , denotes taking the norm, and the upper right superscript 2 denotes squaring; Based on this, the distributed consensus optimization problem is expressed by the formula as:

[0056]

[0057] ;

[0058] where denotes the local estimate of the th submatrix to​​ Local estimates.

[0059] Optionally, the DHEX algorithm adopts the original-dual approach, with the original variables... Initialize as a Gaussian matrix with zero mean and small variance dual variables Initialize it as a zero matrix, let To and Consistent symmetric double random mixture matrix, , express The first in Line number In each iteration, the elements of the column are calculated locally and iteratively updated by combining neighbor information for each subarray.

[0060] In the During the nth iteration, the 1st The local computation of each subarray is expressed by the formula:

[0061] ;

[0062] in, express The complex conjugate matrix, This indicates calculating the gradient;

[0063] In the During the nth iteration, the 1st The iterative update of the subarray's combined neighbor information can be expressed by the formula:

[0064] ;

[0065] ;

[0066] in, , For the two preset step sizes, Indicates the first The original variable output by the next iteration Indicates the first The dual variable output by the next iteration. Indicates the first The original variable output by the next iteration Indicates the first The dual variable output by the next iteration;

[0067] When the number of iterations reaches Stop iterating when a consistent factor is obtained. ,pass Reconstruct the complete signal to obtain the reconstructed signal. .

[0068] Optionally, the reconstructed signals are processed by using a two-dimensional subspace method to obtain a direction of arrival estimation value, specifically: on the basis of the reconstructed signals , a direction of arrival estimation is performed by using a MUSIC algorithm or an ESPRIT algorithm to obtain a direction of arrival estimation value.

[0069] An embodiment of the present application proposes a distributed single-shot direction of arrival estimation method based on two-dimensional Hankel matrix completion, which overcomes the problems of dependence on multiple-shot samples, the need for centralized processing and insufficient accuracy in the case of data loss in the traditional direction of arrival estimation technology, and has the following beneficial effects.

[0070] First, the estimation accuracy under the single-shot condition is effectively improved. The present application can realize high-precision direction of arrival estimation under the single-shot observation condition by introducing two-dimensional Hankel promotion and low-rank completion operation, avoiding the failure of the traditional method due to insufficient samples.

[0071] Second, the estimation robustness is effectively improved. The present application can complete the randomly missing or sparsely sampled data by using the low-rank characteristic of the Hankel matrix, thereby ensuring the integrity and reliability of the estimation.

[0072] Third, the high communication and computation overhead of centralized processing is avoided. The present application selects a distributed processing architecture, and each subarray only needs to use its own local data and exchange a small amount of low-dimensional information with the neighbor subarray to realize collaborative processing, without the need to transmit all the original observation data, thereby reducing the communication burden and system complexity.

[0073] Fourth, the consistency and convergence of distributed processing are ensured. The present application can ensure that all local estimates of the subarrays converge to a consistent solution under the distributed processing architecture by introducing the DHEX distributed consistency optimization algorithm, thereby realizing the performance comparable to the centralized method without relying on the central node. BRIEF DESCRIPTION OF DRAWINGS

[0074] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the related art, the drawings needed to be used in the description of the embodiments of the present application or the related art will be briefly introduced. Obviously, the following drawings are only some embodiments of the present application, and other drawings can be obtained by those skilled in the art without creative labor on the basis of these drawings. The drawings described herein are only used to explain the present application and do not limit the present application.

[0075] Figure 1 is a flowchart of a distributed single-shot direction of arrival estimation method based on two-dimensional Hankel matrix completion provided in an embodiment of the present application;

[0076] Figure 2 is a schematic diagram of the structure and working principle of a distributed linear array system composed of multiple sub-arrays provided in an embodiment of the present application;

[0077] Figure 3 is a schematic diagram of the change of RMSE corresponding to different algorithms with SNR provided in an embodiment of the present application;

[0078] Figure 4 is a schematic diagram of the change of RMSE corresponding to different algorithms with sampling rate provided in an embodiment of the present application;

[0079] Figure 5 is a structural schematic diagram of a distributed single-snapshot direction of arrival estimation system based on two-dimensional Hankel matrix completion provided in another embodiment of the present application;

[0080] Figure 6 is a structural schematic diagram of an electronic device provided in another embodiment of the present application. DETAILED DESCRIPTION

[0081] In order to make the purpose, technical solutions and advantages of the embodiments of the present application clearer, the embodiments of the present application will be described in detail below with reference to the drawings. Those skilled in the art can understand that in the embodiments of the present application, many technical details are proposed in order to make the readers better understand. However, the technical solutions claimed by the present application can be implemented even without these technical details and various changes and modifications based on the following embodiments. The following embodiments are divided for the convenience of description, and should not constitute any limitation on the specific implementation of the present application. The following embodiments can be combined and referenced with each other without contradiction.

[0082] An embodiment of the present application proposes a distributed single-snapshot direction of arrival estimation method based on two-dimensional Hankel matrix completion, which is suitable for a distributed linear array system composed of multiple sub-arrays. The implementation details of the distributed single-snapshot direction of arrival estimation method based on two-dimensional Hankel matrix completion proposed in the embodiment will be described below. The following content is only the implementation details provided for the convenience of understanding, and is not essential for implementing the present solution.

[0083] The specific process of the distributed single-snapshot direction of arrival estimation method based on two-dimensional Hankel matrix completion proposed in the embodiment can be as shown in Figure 1 , which includes:

[0084] Step 11, obtaining the signals received by each sensor of each sub-array and stacking them into a global data matrix, and obtaining an under-sampling observation signal containing additive noise based on an element sampling operator and the global data matrix.

[0085] In a specific implementation, when performing distributed single-snapshot direction of arrival estimation, first, the signals received by each sensor of each subarray need to be obtained and stacked into a global data matrix, and then, based on an element-wise sampling operator and the global data matrix, an under-sampled observation signal containing additive noise is obtained.

[0086] With the increasing demand for low-altitude airspace management and public safety, the positioning and monitoring of unmanned aerial vehicles (UAVs) have become a key issue that needs to be addressed. Traditional UAV positioning relies on radar or satellite navigation systems, but in complex environments such as urban canyons, electromagnetic interference areas, or GNSS-limited scenarios, these methods cannot work stably.

[0087] Under this background, the direction of arrival estimation technology of distributed linear array systems has gradually attracted attention. The basic idea is to deploy multiple subarrays in cities, borders, or key areas. These subarrays are approximately uniformly distributed in space and form a distributed observation network through limited neighbor communication. As shown in FIG. 1, when an unmanned aerial vehicle emits or reflects electromagnetic signals, different subarrays can simultaneously obtain snapshot data of the signals. Through distributed direction of arrival estimation, the incident direction of the unmanned aerial vehicle signal can be calculated, thereby realizing the joint positioning of the unmanned aerial vehicle. Figure 2

[0088] A distributed linear array system is usually composed of multiple subarrays, each containing several sensors. After the far-field signal is received by each sensor, it can be represented as a global observation matrix as a whole. However, in actual applications, due to factors such as hardware failure, limited observation time, or intentional sparse sampling, the obtained observation matrix is incomplete. Therefore, an observation index set and a corresponding sampling operator are usually introduced to retain the signal only at the sampled positions, thereby forming noisy under-sampled observation data (under-sampled observation signal).

[0089] Consider a distributed linear array system composed of uniform linear subarrays, each containing sensors, and are integers greater than 1, the array element spacing is denoted as , the displacement of adjacent subarrays is denoted as , is not necessarily an integer multiple of , then the position of the sensor of the subarray can be represented by the formula:

[0090] ;

[0091] where is the sensor of the ​The location of each sensor , .

[0092] Assume there are a total of A narrowband far-field wave source, denoted as the... A narrowband far-field wave source has a global complex amplitude. The global complex amplitude is determined solely by the narrowband far-field source itself and remains consistent across all subarrays.

[0093] Based on the above scenario, in the absence of noise, the first The first of the sub-arrays The signal received by a sensor can be expressed by the formula:

[0094] ;

[0095] in, For the first The first of the sub-arrays The signal received by the sensor For wavelength, The unknown first DOA of a narrowband far-field source It represents the imaginary unit.

[0096] When stacking, first place the first... All of the individual formations The signals received by each sensor are stacked into an observation vector. Then all The observation vectors corresponding to each subarray are stacked to form a global data matrix. .

[0097] in, , upper right corner mark This indicates the transpose operation. , The dimension is The complex field of .

[0098] In practical applications, due to hardware failures, time constraints, or intentional sparse sampling to reduce complexity, the global data matrix... It is often incomplete; therefore, we need to base it on element-wise sampling operators and the global data matrix. This yields an undersampled observation signal containing additive noise.

[0099] First, define For the observation index set, At the same time, define For element-wise sampling operators, By formula:

[0100] ;

[0101] wherein, denotes the iteration variable, denotes the local estimate of the th sensor pair of the th subarray.

[0102] On this basis, the undersampled observation signal containing additive noise is obtained based on the element-wise sampling operator and the global data matrix by the following formula:

[0103] ;

[0104] wherein, denotes the additive noise, denotes the undersampled observation signal, , .

[0105] To define as the local observation index set of the th subarray, , , denotes the identity, and are consistent in dimension, but only the entries corresponding to the th row are retained, and is defined as the local element-wise sampling operator of the th subarray, which is expressed by the formula:

[0106] .

[0107] Obviously, , , based on which, the local undersampled observation signal of the th subarray can be expressed by the formula:

[0108] ;

[0109] wherein, denotes that the th subarray can only access the local undersampled observation signal, and the th subarray can only access .

[0110] The target of the embodiment is to estimate the directions of arrival of each signal source by using the undersampled observation signal. ​

[0111] Step 12, two-dimensional Hankel low-rank promotion is performed on the global data matrix to construct a two-dimensional Hankel matrix.

[0112] In a specific implementation, after obtaining the global data matrix and the under-sampling observation signal, two-dimensional Hankel low-rank promotion is performed on the global data matrix to construct a two-dimensional Hankel matrix.

[0113] It can be understood that the element in the i-th row and the j-th column of the global data matrix satisfies:

[0114] ;

[0115] wherein, , , .

[0116] Based on this, the Vandermonde vector and are defined as:

[0117] ;

[0118] ;

[0119] wherein, , , represents a complex field with a dimension of , represents a complex field with a dimension of .

[0120] Based on and , the global data matrix is expressed as , , represents the rank of .

[0121] However, in practice, if or is not large enough, the “low-rank property” of may not be sufficient. Therefore, we need to further find a more advantageous structure, i.e., two-dimensional Hankel promotion, to facilitate matrix completion.

[0122] , , , , so that​​​ , ;

[0123] Definition of one-dimensional Hankel operator along the sensor dimension , , denotes the complex field of dimension . For a vector of length , satisfies .

[0124] The global data matrix is applied row-wise , resulting in small block matrices. The th block matrix corresponding to the th row of the global data matrix , ;

[0125] Based on all small block matrices, a two-dimensional Hankel matrix is constructed along the matrix dimension, denoted as:

[0126] ;

[0127] where , denotes the complex field of dimension .

[0128] Next, it is proven that has low-rank property.

[0129] Lemma 1, .

[0130] It is proven that for a single source a , th row of behaves as .

[0131] Therefore, .

[0132] where , , and are intermediate variables.

[0133] Substituting into the matrix form of , we get:

[0134] ;

[0135] wherein, , , and are intermediate variables, denotes taking the Kronecker product.

[0136] By using the property of Kronecker product, we can get:

[0137] .

[0138] The linear by Hankel lifting can be obtained:

[0139] ;

[0140] Therefore, we can get, .

[0141] According to Lemma 1, has low rank, and the low rank completion consistent with the observation can be performed accordingly.

[0142] Step 13, based on the undersampling observation signal, the calculation of the local cost function is performed, the completion problem of the two-dimensional Hankel matrix is converted into a distributed consistency optimization problem, the consistency iteration is realized through the DHEX algorithm, the global low rank solution is obtained, and the complete signal reconstruction is completed, and the reconstructed signal is obtained.

[0143] In the specific implementation, after obtaining the two-dimensional Hankel matrix, the calculation of the local cost function can be performed based on the undersampling observation signal, the completion problem of the two-dimensional Hankel matrix is converted into a distributed consistency optimization problem, the consistency iteration is realized through the DHEX algorithm, the global low rank solution is obtained, and the complete signal reconstruction is completed, and the reconstructed signal is obtained. The DHEX algorithm is a distributed two-dimensional Hankel matrix completion algorithm designed based on the classic distributed algorithm EXTRA (An Exact First-Order Algorithm for Decentralized Consensus Optimization), which is used to solve the distributed consistency optimization problem.

[0144] In the process of signal reconstruction, first, the communication modeling is performed, and the communication network between the subarrays is set as an undirected graph , wherein, is a set of subarrays, is a communication edge, each subarray can only exchange data with its neighbor subarray in the optimization process to meet the distributed constraint.

[0145] Next, we will perform distributed modeling based on two-dimensional Hankel matrix completion, taking... and Let it be an odd number. and Satisfy constraints: , The purpose of this constraint is to enhance symmetry and reduce communication overhead.

[0146] At this point, the two-dimensional Hankel matrix is ​​increased. For square array , , The dimension is The complex field, , It has transpose property.

[0147] Using the symmetric low-rank decomposition method, Decomposed into parameterized , , The dimension is The complex field of .

[0148] Definition of the first The local cost function for each subarray is:

[0149] ;

[0150] in, Indicates the first The local cost of each unit Indicates the first Individual pairs Local estimates, Sampling rate, , For identity operators, , for Accompanying, It is an invertible linear weighted operator, indicated by the superscript. This represents the inverse operation, for , satisfy , and They are respectively and That is, the first in the matrix The diagonal line (from top left to bottom right) and the first The length of the diagonal line (from the top right to the bottom left). Indicates taking Norm, with a superscript 2 indicating that it is squared.

[0151] The first guarantee is consistent with the data, and the second guarantee is the Hankel structure.

[0152] Based on this, the distributed consensus optimization problem can be expressed by the following formula:

[0153] ;

[0154] in, Indicates the first Individual pairs Local estimates.

[0155] Finally, the algorithm is used to solve the problem, specifically through the DHEX algorithm to achieve consensus iteration, obtain the global low-rank solution, and reconstruct the complete signal, thus obtaining the reconstructed signal. The DHEX algorithm adopts a primal-dual approach, with the original variables... Initialize as a Gaussian matrix with zero mean and small variance dual variables Initialize it as a zero matrix, let To and Consistent symmetric double random mixture matrix, , express The first in Line number In each iteration, the elements of the column are calculated locally and updated iteratively by combining neighbor information.

[0156] In the During the nth iteration, the 1st The local computation of each subarray is expressed by the formula:

[0157] ;

[0158] in, express The complex conjugate matrix, This indicates calculating the gradient.

[0159] In the During the nth iteration, the 1st The iterative update of the subarray's combined neighbor information can be expressed by the formula:

[0160] ;

[0161] ;

[0162] in, , For the two preset step sizes, Indicates the first the original variable output in the i-th iteration, the dual variable output in the i-th iteration, the original variable output in the i-th iteration, the dual variable output in the i-th iteration, the original variable output in the i-th iteration, the dual variable output in the i-th iteration, the dual variable output in the i-th iteration.

[0163] When the number of iterations reaches , the iteration is stopped, a consistent factor is obtained, the complete signal is reconstructed by , and a reconstructed signal is obtained.

[0164] Step 14: The reconstructed signal is processed by using a two-dimensional subspace method to obtain a direction of arrival estimation value.

[0165] In a specific implementation, after the reconstructed signal is obtained, the two-dimensional subspace method can be used to process the reconstructed signal, so as to obtain the direction of arrival estimation value.

[0166] In an example, on the basis of the reconstructed signal , the MUSIC algorithm or the ESPRIT algorithm is used for direction of arrival estimation to obtain the direction of arrival estimation value. It should be noted that usually only the estimation related to the frequency in the array is used, because the displacement between the subarrays is usually greater than half the wavelength, and if is used, phase ambiguity may occur.

[0167] The distributed single-shot direction of arrival estimation method based on two-dimensional Hankel matrix completion proposed in the embodiment overcomes the problems in the traditional direction of arrival estimation technology, such as dependence on multiple-shot samples, the need for centralized processing, and insufficient accuracy in the case of data loss, and has the following beneficial effects.

[0168] First, the estimation accuracy under the single-shot condition is effectively improved. The two-dimensional Hankel promotion and low-rank completion operation are introduced in the embodiment, so that high-precision direction of arrival estimation can be realized under the single-shot observation condition, and the failure caused by insufficient samples in the traditional method is avoided.

[0169] Second, the estimation robustness is effectively improved. The low-rank characteristic of the Hankel matrix is used in the embodiment to complete the data sampled randomly or sparsely, so as to ensure the integrity and reliability of the estimation.

[0170] Third, the high communication and computation overhead of centralized processing is avoided. In this embodiment, a distributed processing architecture is selected, each subarray only needs to utilize its local data and exchange a small amount of low-dimensional information with neighbor subarrays to achieve collaborative processing, without transmitting all raw observation data, thereby reducing the communication burden and system complexity.

[0171] Fourth, the consistency and convergence of distributed processing are ensured. In this embodiment, the DHEX distributed consistency optimization algorithm is introduced, which can ensure that all local estimates of the subarrays converge to a consistent solution under the distributed processing architecture, thereby achieving performance comparable to centralized methods without relying on a central node.

[0172] The step division of the above methods is only for the purpose of clear description, and in actual implementation, one step can be combined or some steps can be divided into multiple steps, as long as the same logical relationship is included, which is within the protection scope of the present application. Irrelevant modifications or irrelevant designs are added to the algorithm or the flow, but the core design of the algorithm and the flow is not changed, which is within the protection scope of the present application.

[0173] In one embodiment, in order to verify the superior performance of the DHEX algorithm under distributed, single-snapshot DOA estimation, relevant simulation experiments are performed, and the performance of the DHEX algorithm is compared with that of three other algorithms, which are: Zero-fill, directly setting the missing entries of the observation matrix to zero; centralized X-domain matrix completion (CXMC), performing low-rank matrix completion in a centralized environment; and centralized two-dimensional Hankel matrix completion (CHMC), using a centralized two-dimensional Hankel completion based on spectral gradient.

[0174] The following settings are unified. The array element spacing , the subarray size , the number of subarrays , the displacement of adjacent subarrays , the number of sources , the true value DOA, i.e. is , and the amplitude of each source is a complex Gaussian random variable. The step size of CHMC is , the step size of DHEX is , , the maximum number of iterations is , the communication graph is a ring graph, and the performance index is the root mean square error (RMSE). The RMSE is expressed by the formula:

[0175] ;

[0176] wherein is the estimate of .

[0177] In the SNR (Signal-to-Noise Ratio) experiment, the sampling rate is 0 . 6, SNR from 0dB to 40dB, step 5dB. 100 independent Monte Carlo experiments are conducted under each SNR to evaluate different algorithms. Figure 3 It is shown that CHMC and DHEX are significantly better than Zero-fill and CXMC in the whole SNR range, especially in the low SNR region. In addition, although DHEX is a distributed algorithm, its performance is very close to CHMC, which verifies the effectiveness of the distributed framework of the present application.

[0178] In the sampling rate experiment, SNR is fixed at 20dB, sampling rate from 0 . 4 to 0 . 9, step 0 . 1. Similarly, 100 Monte Carlo experiments are conducted for each sampling rate to evaluate the robustness under different missing levels. It can be seen from Figure 4 that under different sampling rates, CHMC and DHEX always achieve the lowest RMSE, Zero-fill error is the largest, and although CXMC is improved compared with Zero-fill, there is a significant gap with CHMC and DHEX at low sampling rate.

[0179] Another embodiment of the present application proposes a distributed single-snapshot direction of arrival estimation device based on two-dimensional Hankel matrix completion, which is suitable for a distributed linear array system composed of multiple sub-arrays. The details of the distributed single-snapshot direction of arrival estimation device based on two-dimensional Hankel matrix completion proposed in this embodiment will be described below. The following content is only provided for the implementation details for easy understanding, and is not necessary for implementing this embodiment.

[0180] Figure 5 is a structure diagram of the distributed single-snapshot direction of arrival estimation device based on two-dimensional Hankel matrix completion proposed in this embodiment, which comprises: an undersampled observation signal acquisition module 21, a two-dimensional Hankel low-rank lifting module 22, a signal reconstruction module 23 and an estimation execution module 24.

[0181] The undersampled observation signal acquisition module 21 is used to acquire the signals received by each sensor of each sub-array and stack them into a global data matrix, and based on an element sampling operator and the global data matrix, an undersampled observation signal containing additive noise is obtained.

[0182] The two-dimensional Hankel low-rank lifting module 22 is used to perform two-dimensional Hankel low-rank lifting on the global data matrix to construct a two-dimensional Hankel matrix.

[0183] The signal reconstruction module 23 is used to calculate the local cost function based on the undersampled observation signal. It transforms the problem of completing the two-dimensional Hankel matrix into a distributed consensus optimization problem. The consensus iteration is achieved through the DHEX algorithm to obtain the global low-rank solution and complete the reconstruction of the complete signal, thus obtaining the reconstructed signal.

[0184] The estimation execution module 24 is used to process the reconstructed signal using a two-dimensional subspace method to obtain the direction of arrival estimate.

[0185] It is worth mentioning that all modules and units involved in this embodiment are logical modules. In practical applications, a logical unit can be a physical unit, a part of a physical unit, or a combination of multiple physical units. Furthermore, to highlight the innovative aspects of this application, this embodiment does not introduce units that are not closely related to solving the technical problems proposed in this application; however, this does not mean that other units do not exist in this embodiment.

[0186] It is not difficult to see that this embodiment is a system embodiment corresponding to the above method embodiments, and this embodiment can be implemented in conjunction with the above method embodiments. The relevant technical details and technical effects mentioned in the above method embodiments are still valid in this embodiment, and will not be repeated here to reduce repetition. Accordingly, the relevant technical details mentioned in this embodiment can also be applied to the above method embodiments.

[0187] Another embodiment of this application provides an electronic device, such as Figure 6 As shown, it includes a processor 31 and a memory 32. The memory 32 stores instructions that the processor 31 can execute. When the processor 31 is configured to execute the instructions, the electronic device can implement a distributed single-snapshot direction-of-arrival estimation method based on two-dimensional Hankel matrix completion as described in the above method embodiment.

[0188] The memory and processor are connected via a bus, which includes any number of interconnecting buses and bridges, connecting various circuits of one or more processors and memories. The bus can also connect various other circuits such as peripheral devices, voltage regulators, and power management circuits, which are well known in the art and will not be described further herein. The bus interface provides an interface between the bus and the transceiver. The transceiver can be a single element or multiple elements, such as multiple receivers and transmitters, providing a unit for communicating with various other devices over a transmission medium.

[0189] The processor is responsible for managing the bus and general processing, and can also provide various functions, including timing, peripheral interface, voltage regulation, power management, and other control functions. The memory can be used to store data used by the processor in performing operations.

[0190] Another embodiment of the present application provides a computer readable storage medium, wherein a computer program is stored in the computer readable storage medium, and the computer program, when executed by a processor, implements a distributed single fast time of arrival direction estimation method based on two-dimensional Hankel matrix completion.

[0191] That is, a person skilled in the art can understand that all or part of the steps in the above method embodiments can be completed by programs instructing relevant hardware, and the programs are stored in a storage medium and include a plurality of instructions for causing a device (such as a single-chip microcomputer, a chip, etc.) or a processor to execute all or part of the steps of the method described in the method embodiments of the present application. The foregoing storage medium includes a U disk, a mobile hard disk, a read-only memory, a random access memory, a magnetic disk or an optical disk, and various storage media that can store program codes.

[0192] A person skilled in the art can understand that the above embodiments are specific embodiments of the present application, and in actual application, various changes can be made in form and details without departing from the spirit and scope of the present application. For those skilled in the art, a number of improvements and refinements can be made without departing from the principles of the present application, and these improvements and refinements are also considered to be within the protection scope of the present application.

Claims

1. A distributed single-snapshot direction of arrival estimation method based on two-dimensional Hankel matrix completion, applicable to a distributed linear array system composed of multiple sub-arrays, characterized in that, The method comprises: acquiring signals received by sensors of each subarray and stacking into a global data matrix, obtaining an under-sampling observation signal containing additive noise based on an element-wise sampling operator and the global data matrix; performing two-dimensional Hankel low-rank promotion on the global data matrix to construct a two-dimensional Hankel matrix; calculating a local cost function based on the under-sampling observation signal, converting a completion problem of the two-dimensional Hankel matrix into a distributed consistency optimization problem, realizing consistency iteration through a DHEX algorithm, obtaining a global low-rank solution and completing reconstruction of a complete signal to obtain a reconstructed signal; processing the reconstructed signal by using a two-dimensional subspace method to obtain a direction of arrival estimation value; The distributed linear array system is composed of uniform linear sub-arrays, each of which contains sensors, and are integers greater than 1, the element spacing is denoted as , the displacement of adjacent sub-arrays is denoted as , and the position of the th sensor in the th sub-array is expressed by the formula ； wherein is the position of the th sensor of the th subarray, , ; acquiring signals received by sensors of each subarray and stacking into a global data matrix, comprising: There are a total of narrowband far-field wave sources in the space The first narrowband far-field wave source has a global complex amplitude The global complex amplitude is determined only by the narrowband far-field wave source itself and remains consistent on all subarrays In the absence of noise, the signal received by the jth sensor of the ith subarray is given by: The signal received by the jth sensor of the ith subarray is given by:​ ； wherein is the DOA of the th sensor of the th subarray, is the wavelength, denotes the DOA of the th narrowband far-field wave source, denotes the imaginary unit; The signals received by all the sensors of the first subarray are stacked into an observation vector , and the observation vectors corresponding to all the subarrays are stacked into a global data matrix ;​​​ wherein , , the upper right corner denotes the transpose operation, , denotes the complex field of dimension . Definitions is defined as the set of indices, , is defined as is defined as the element-wise sampling operator, is expressed by the formula ； in, Represents the iteration variable. Indicates the first The first of the sub-arrays Each sensor pair The local estimate; obtaining an under-sampling observation signal containing additive noise based on an element-wise sampling operator and the global data matrix through the following formula: ； wherein represents an additive noise, represents an undersampled observation signal, , ; Definition The local observation index set of the th sub-array, , , , denotes the identity, and has the same dimension as, but only keeps the entries corresponding to the th row, and defines the local element-wise sampling operator of the th sub-array, , The local observation index set of the th sub-array, , is expressed by the formula: ； No. The local undersampled observation signal of each subarray can be expressed by the formula: ； wherein, represents the subarray can only access the local under-sampled observation signal, , the subarray can only access .

2. The distributed one-shot direction of arrival estimation method based on two-dimensional Hankel matrix completion according to claim 1, characterized in that, global data matrix of the first row of the first column of the first element satisfies: ； wherein , , ; Based on this, the Vandermonde vector and is expressed as: ； ； wherein , , denotes a complex field of dimension , denotes a complex field of dimension . based on and global data matrix denoted as , , denotes rank of performing two-dimensional Hankel low-rank promotion on the global data matrix to construct a two-dimensional Hankel matrix, comprising: integer , , , so that , ; Defining a one-dimensional Hankel operator along a sensor dimension , , denotes the complex field of dimension , for a vector of length , satisfies ; to the global data matrix is applied row by row , resulting in sub-matrices, the sub-matrix corresponding to the th row of the global data matrix is , ; Based on all the small block matrices, a two-dimensional Hankel matrix is constructed along the matrix dimension is represented as: ； wherein , denotes the complex field of dimension .

3. The method of claim 2, wherein, calculating a local cost function based on the under-sampling observation signal, converting a completion problem of the two-dimensional Hankel matrix into a distributed consistency optimization problem, comprising: The communication modeling is performed, and the communication network between the subarrays is a undirected graph , wherein, is a set of subarrays, is a communication edge, each subarray can only exchange data with its neighbor subarrays in the optimization process to meet the distributed constraints; Distributed modeling based on two-dimensional Hankel matrix completion is performed, taking and is odd, let and satisfy the constraints: , ; Elevating two-dimensional Hankel matrices is a square matrix , , denotes a complex field of dimension ;​ Using a symmetric low-rank decomposition method, the matrix is decomposed into a parameterized , , denotes the complex field of dimension . The local cost function for the first subarray is defined as: ； wherein, denotes the local cost of the th subarray, denotes the local estimate of the th subarray pair , is the sampling rate, , is the identity operator, , is the adjoint of , is a reversible linear weighting operator, the upper right index denotes the inverse operation, for , satisfies , and are respectively and , denotes the taking norm, the upper right index 2 denotes the squaring; Based on this, the distributed consistency optimization problem is represented by the formula as: ； wherein, denotes the local estimate of the th subarray pair .

4. The distributed one-shot DOA estimation method based on two-dimensional Hankel matrix completion according to claim 3, characterized in that, The DHEX algorithm uses a primitive-dual approach, with the original variables... Initialize as a Gaussian matrix with zero mean and small variance dual variables Initialize it as a zero matrix, let To and Consistent symmetric double random mixture matrix, , express The first in Line number In each iteration, the elements of the column are calculated locally and iteratively updated by combining neighbor information for each subarray. At the first iteration, the local computation of the first subarray is expressed by the formula: At the second iteration, the local computation of the second subarray is expressed by the formula: = 1, 2, 3,..., N - 1 ； wherein represents the complex conjugate matrix of denotes the gradient; At the first iteration, the combined neighbor information of the first subarray is updated by the formula: At the second iteration, the combined neighbor information of the first subarray is updated by the formula: the first iteration, the combined neighbor information of the first subarray is updated by the formula: ； ； in, , For the two preset step sizes, Indicates the first The original variable output by the next iteration Indicates the first The dual variable output by the next iteration. Indicates the first The original variable output by the next iteration Indicates the first The dual variable output by the next iteration; The iterations are stopped when the number of iterations reaches and the consistent factors are obtained. The complete signal is reconstructed by and the reconstructed signal is obtained.

5. The distributed one-shot DOA estimation method based on two-dimensional Hankel matrix completion according to claim 4, characterized in that, The two-dimensional subspace method is used to process the reconstructed signals to obtain the direction of arrival estimation value, specifically: on the basis of the reconstructed signals , the MUSIC algorithm or the ESPRIT algorithm is used for direction of arrival estimation to obtain the direction of arrival estimation value.

6. A device for distributed single-snapshot direction of arrival estimation based on two-dimensional Hankel matrix completion, which is suitable for a distributed linear array system composed of multiple sub-arrays, and is implemented based on a method for distributed single-snapshot direction of arrival estimation based on two-dimensional Hankel matrix completion according to any one of claims 1 to 5, characterized in that, The device comprises: an under-sampling observation signal acquisition module configured to acquire signals received by sensors of each subarray and stack into a global data matrix, and obtain an under-sampling observation signal containing additive noise based on an element-wise sampling operator and the global data matrix; a two-dimensional Hankel low-rank promotion module configured to perform two-dimensional Hankel low-rank promotion on the global data matrix to construct a two-dimensional Hankel matrix; a signal reconstruction module configured to calculate a local cost function based on the under-sampling observation signal, convert a completion problem of the two-dimensional Hankel matrix into a distributed consistency optimization problem, realize consistency iteration through a DHEX algorithm, obtain a global low-rank solution, complete reconstruction of a complete signal, and obtain a reconstructed signal; an estimation execution module configured to process the reconstructed signal by using a two-dimensional subspace method to obtain a direction of arrival estimation value.

7. An electronic device, comprising: comprise: a processor and a memory, wherein the memory stores instructions executable by the processor, and the processor is configured to execute the instructions so that the electronic device can implement a distributed single-snapshot direction of arrival estimation method based on two-dimensional Hankel matrix completion according to any one of claims 1 to 5.

8. A computer readable storage medium storing a computer program, characterized in that, The computer program, when executed by a processor, can implement a distributed single-snapshot direction of arrival estimation method based on two-dimensional Hankel matrix completion according to any one of claims 1 to 5.

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